Spinoriality of orthogonal representations of $\mathrm{GL}_n(\mathbb{F}_q)$

Rohit Joshi and Steven Spallone

Vol. 311 (2021), No. 2, 369–383

DOI: 10.2140/pjm.2021.311.369

Abstract

We determine which orthogonal representations $V$ of ${GL}_n\left({\mathbb{𝔽}}_q\right)$ lift to the double cover $Pin\left(V\right)$ of the orthogonal group $O\left(V\right)$. We cover all $n\ge 1$ and prime powers $q$, except for $\left(n,q\right)=\left(3,4\right)$.

Keywords

orthogonal representations, spinoriality, finite groups of Lie type

Mathematical Subject Classification

Primary: 20G05

Secondary: 20G40

Milestones

Received: 15 August 2020

Revised: 29 December 2020

Accepted: 13 January 2021

Published: 31 July 2021